One of the simplest concepts in finance is the relationship between risk and reward. Indeed, much of modern financial theory is based on the notion that holders of riskier investments will invariably be compensated with greater return for bearing said risk. However, what does the empirical data suggest? Does it bear it out in practice? I took up a simple study to try and answer this question and found that the data does seem to suggest there is some degree of compensation for bearing additional risk (at least in U.S. large cap equities). However, there does appear to be a threshold where the fundamental risk-return relationship breaks down and actually goes the other direction, i.e., at high levels of volatility you actually get compensated with less return per each unit of new risk assumed.

The chart above is breaking out the returns on the S&P 500 into 5 quintiles based on a rolling 260-day standard deviation of daily price returns for each stock in the index. You are looking at forward returns on the quintiles that are rebalanced semiannually and cap-weighted like the overall index for sake of consistency. In other words, every six months I’m looking at the current 260-day standard deviation of daily price return and reclassifying all stocks in the index into one of the five quintiles of roughly 100 stocks each. Quintile 1 represents the 100 least volatile stocks in the S&P 500 on the basis of 260-day standard deviation while quintile 5 represents the most volatile 100 stocks in the index. As you can see, quintiles 1 through 4 appear to behave somewhat along the lines of fundamental risk-reward concepts, with quintile 4 giving greater return while at greater risk. However, quintile 5 is a complete deviation from the group, giving the smallest amount of return and the highest degree of risk.

This chart gives the relationship between annualized return and standard deviation on each of the risk quintiles I described above. I also superimposed a non-scientific generalized curve that shows what basic financial theory might expect from each of the risk quintiles in the S&P 500. As you can see, quintile 5 represents a complete break from what academic thinking of asset returns would assume. I find this particularly interesting as it represents such a large deviation from conventional thought.

So I guess the question becomes why do the highest risk members of the S&P 500 do so poorly on a risk-return comparison and stand in contrast to common academic theory? To me the answer is probably not in the fundamentals of high risk stocks but in the mathematics of compound returns. Which is ironic, because one of the first things you learn in any finance class is the concept that geometric average (how you calculate annualized returns) declines as you increase the variability of the member components in a chain-linked return series. I think academic finance reconciles the mathematical aspects of geometric returns with theoretical risk-return relationships by arbitrarily slapping on risk premia into expected returns of higher risk assets. So even though the variability of returns is greater on high risk assets, their expected drift is more strongly positive over time, more than offsetting the reduction in annualized return from return variability. If that is the case, then the above observation may be that whatever these risk premia for high risk assets may be, they are either overstated or entirely nonexistent.

I’ve reproduced a similar analysis for U.S. small cap equities as represented by the Russell 2000 above. What I find interesting here is that the risk-return relationship seems to break down entirely as you go from the lowest risk quintile to the highest risk quintile. This is an interesting result that I did not expect. My initial thinking when I wrote this post was that higher risk-higher return holds true but up to a point. The data from the Russell 2000 analysis seems to suggest zero benefit on an absolute return basis to holding anything but the least volatile U.S. small cap equities. As you go out to higher volatility quintiles, you are forcing yourself to accept less return and take on more risk.